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Sciences 
Cbrporation 


33  WEST  MAIN  STPEET 

WEBSIk:R,N.Y.  MS80 

(716)  872-4503 


CIHM/ICMH 

Microfiche 

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dernidre  image  de  cheque  microfiche,  selon  le 
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et  de  haut  en  bas,  en  prenant  le  nonr.bre 
d'images  ndcessaire.  Les  diagrammes  suivants 
illustrent  la  m6thode. 


1 

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2 

3 

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4 

5 

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REPRINXED    FF.OM 


-  ;fi\,^j. 


\^^■r 


ANNALS  OF  MATHEMATICS. 


•^-. 


Vol..  X. 


April,  1896. 


No.  4..-V 


ON  THE  SOLUTION  OP  A  CERTAIN  DIFFERENTIAL  EQUATION 
WHICH  PRESENTS  ITSELF  IN  LAPLACE'S  KINETIC  THEORY 
OF  TIDES. 

-    '•    -    By  Mr.  Geoboe  Hebbert  Lino,  New  York,  N.  Y.  -■-•<■■•  > 

[Siibmittad  in  pfxrtinl  fulfilment  of  tbe  requirements  for  .he  degree  of  Doctor  of  Pbilosopliy  iu  tlic 
Faculty  of  Pure  Science  of  Columbift  UniverBity  iu  the  City  of  New  York.] 


,:^i.,\ 


■  'X  '■' 


!j ; 


„.    X 


■<1,.,     ■.  '■■'■' 


'% 


r^^ 


m 


^»^  *■*.-<*,•,*,, 


c 


rp.'.. 
\/.sI ^^f 


ON  THE  SOLUTION  OF  .\  CEIITATN  DIFFERENTIAL  EQUATION 
WHKJH  PRESENTS  ITSELF  IN  LAPLACE'S  KINETIC  THEORY 
OF  TIDES. 

By  Vu.  Gkoiuik  Mkubeiit  IjIN<i,  New  York,  N.  Y. 

^  TABLE  or  CONTENTS. 


0  A  ■ 


Paiik. 


.\it.  1.  Objects  of  the  paper, 


I.  IntboductohV. 
II.   HiHTOKicAi.  Sketch. 


10 
11, 
12, 
111 


14. 
15. 

10. 
17. 

18, 
1'.), 
20, 
21, 
22 
23 
24 


25, 
2fi. 


Origin  of  the  prohleiu, 

Previous  ooiitrilnitioim  to  the  subject,         ..... 
Synopsis  of  previous  contributions 

III.  Laplace's  Treatment. 

General  Outline, 

AppliciitioM,       ......•••• 

Objections  to  the  nietliod, 

lU'iiition  between  coireetion  to  the  e(iuiitioii  luul  error  in  series. 
.\ssnniption  eiinivalent  to  Liipliice's  nssuniptiou, 

IV.  The  Solution  oe  the  Equation. 

(.Ihanioter  of  the  iiitei,'riil.  ...•••••• 

Deduction  of  the  coniph'nieutiiry  fuuetion, •         • 

'I'lie  piirticular  iutegriil 

Properties  of  the  complete  inte^jra!, 

V.  The  Okteiimination  of  the  Constants  fou  Laplace's  Case. 

The  phy-iiciil  (■onilitions, 

Proof  timt  y;  =  0 

VI.    DaUWIN's   I'llESENTATION   OF   Loill)   KeLVIN'h   PROOF   THAT    Li   MU.ST   BE   ZeRO 

Darwin's  argument, 

Discussion  of  Darwin's  proof 

VII.  Application  of  General  Integral  to  Other  Cases. 
Cases  to  l)e  treated 


Polar  i 


Sea  exteudinfi  eqiially  on  l)oth  sides  of  the  ecpiator,     .... 
Sen  liounded  liy  two  pandlels  of  latitude  on  the  same  side  of  the  equator, 

Canal  of  width  2(nyii.i.,'ali>n>,' a  parallel 

Tide  nt  point  distant  .)  from  the  boundary  of  canal,     .... 

Canal  of  negligible  width, 

VIII.  Summary  of  IIesiilts. 

Suiniunrv  of  I  IV, 

Suinnuiry  of  V  VII, 


90 

1)7 
!)7 

!)0 

99 

101 

101 

102 

103 
104 

107 
109 


110 
111 


113 
114 

11.-. 

no 

117 
119 
120 
122 
123 


124 
124 


96  use.      ON  THE  HOLUTION  Ol'  A  CEUrAIN  DIFFEllESTIAI,  KtM'ATION 


I. 

Intuoductory. 

1.  Ohjiu-tn  iif  till'  /)'ij>er.  In  his  discuHsion  of  the  kinetic  theory  of  tides, 
Liii>laco  found  that  tlu!  function  expressiu}j;  the  hei^Iit  of  the  tide  at  ii  '.Mven 
point  due  to  tiie  attraction  of  tlie  disturl)iu},'  hody  satistied  a  certain  ditfereu- 
tial  eiiuation.  l-'indin},'  liiniself  unable  to  ohtain  tlie  ^jfeneral  soUition  of  the 
dilferential  equation,  lie  applied  himself  to  the  discussion  of  several  partimihir 
eases  which  aris(!  when  certain  assumptions  are  made  re<,'ardinj,,'  the  physical 
constitution  of  the  ocean.  Oni;  of  the  cases  lus  treated  was  that  of  tlu!  semi- 
diurnal tide  when  tlu^  dei)th  of  the  ocean  is  supposed  to  be  constant.  In  the 
course  of  his  treatment  of  this  case  certain  considerations  enter  which  have 
given  rise  to  much  discussion.  It  is  pro[)osed  to  devote  some  attention  to  this 
ease,  and  it  is  hoi)ed  to  extend  the  treatmtJiit  of  this  case  so  as  to  include  some 
phases  of  it  not  previously  treated.  While  it  is  generally  conceded  that  the 
facts  in  r(!j^ard  to  the  disputed  point  referred  to,  have  been  made  evident,  yet 
the  methods  of  i)lacing  those  facts  in  evidence  have  been  called  into  question 
by  several  writers  on  the  subject,  and  do  not  appear  to  be  the  most  satisfao 
tory  ones  that  are  available. 

II. 

HisTOKicAL  Sketch. 

2.  Orii/i))  of  the  pruhleni.  As  just  mentioned,  the  subject  to  be  discussed 
was  first  treated  by  Laplace.  His  kinetic  theory  of  tides  is  set  forth  in  the 
Mecaniqne  Celeste,  and  the  part  with  which  we  are  concerned  is  to  be  found 
in  Livre  IV  of  that  work,  his  solution  of  the  differential  ecpiation  being  given 
in  Article  10.  Considerable  time  had  elapsed  between  his  Hrst  di.scussion  of 
the  subject  and  the  publication  of  his  great  work.  The  earliest  presentation 
of  his  treatment  of  the  subject  was  contained  in  a  memoir*  presented  to  the 
Academie  des  Sciences,  and  contained  in  Tome  IX  of  the  (Euvres  de  Laplace. 
He  has  sought  a  solution  of  the  ecjuation  in  the  form  of  a  series  of  positive 
entire  powers,  and  has  made  use  of  a  certain  infinite  continued  fraction  in  the 
evaluatijii  of  one  of  the  coefHcients  of  the  series.  The  correctness  of  the  value 
found  by  his  method  has  been  questioned.  As  the  solution  in  the  series  form 
was  made  the  basis  of  his  calculations,  it  was  of  great  importance  that  no  mis- 
take should  be  made  in  the  determiuat:  m  of  the  coelHcieuts,  and  more  esjie- 
cially  in  the  determination  of  those  occurring  early  iu  the  series. 

*  Keiheiches  Kiir  phisieurs  poiuts  du  Systeme  du  Mouae.     Meraoire-  <lij  rAoadumie  royiile  do 
Puris,  uuut'e  ITTo  C. 


1 


T 


W  IIKII  I'ltESENTS  rrsFT.F  IN  I,AI'LA('K's  KINETIC  TIIEOI      OF  TIHEH. 


'.•7 


3.  J'rerioiiti  contrihiitiunx  to  the  Nn/>}'Tt.  fu  his  oiirly  -lUMiioir  liapliice 
luiH  f,'oiio  sonuswlmt  nioro  into  detail,  ami  tlio  mothod  by  wliicli  lio  dctciiiiiiKHl 
tlm  value  of  the  eoisrtitiiisiit  is  eleavly  shown.  In  tiio  hiter  work  he  has  omitted 
a  Rieat  part  of  the  exphmation,  and  has  contented  himself  with  exjiressin^'  the 
quantity  in  tlie  form  of  Uie  eoiitinued  fnietion  to  whieli  reference  has  heen 
made.  The  later  presentation  of  the  sul>jeet  has  heen  the  more  aeeessiMe  of 
tlie  two,  and  on  it  all  the  later  writers  appear  to  have  based  their  remarks  eon- 
cerninf;  [.aplace's  method,  wliile  the  orij^inal  ])resentation  has  been  overlooked. 
Attentio'i  has  been  called  to  it  by  Prof.  Lamb  in  his  recent  work  on  Hydro- 
dynamics, and  to  him  seems  to  be  due  the  it'discovery,  so  to  speak,  of  the 
memoir.  Laplace's  evaluation  of  the  coetHcient  was  objected  to  bs  Sir  G.  13. 
Airy,*  and  later  the  same  objection  was  made  by  Mr.  William  Kerrel.t  A 
defence  of  Lajdace  w;ts  made  by  Lord  K<^lvin  in  the  rhilosopliical  Ma^'a/.ine 
for  September,  1875.  The  October  number  of  the  same  journal  for  1875  con- 
tains a  note  writttiii  by  Airy  in  which  ho  reatHrms  his  objections  to  Lajjlace's 
result,  and  appears  not  to  re<,'ard  Kelvin's  reasoninj,'  as  convincing?.  The  num- 
ber of  this  journal  for  March,  1>S7(),  contains  a  reply  by  Ferrel  to  the  arguments 
of  Lord  Kelvin.  Prof.  G.  H.  Darwin  in  the  Encyclopedia  Britannica:|:  gives  in 
more  detail  Lord  Kelvin's  argument.  His  treatment  of  the  subject  may  also 
bo  found  in  IJasset's  Hydrodynamics,  Vol.  II,  and  Basset  has  briefly  referred 
to  the  subject  in  a  foot  note.jj  The  latest  contributions  to  the  subject  are 
believed  to  be  the  two  papers  by  Ferrel  which  appear  in  Volumes  9  and  10  of 
Gould's  Astronomical  Journal.  The  latter  of  the  two  papers  may  also  be  found 
in  the  collection  of  papers  ||  on  the  "  Mechanics  of  the  Atmosphere  "  edited  by 
Prof.  Cleveland  Abbe,  lleference  may  also  be  made  to  Professor  Lamb's 
Hydrodynamics,  in  which  attention  is  called  to  Laplace's  original  memoir. 

4.  S)//ii)/iniK  iif  pri'i'hivti  niiitrihiitidiin.  Before  treating  the  problem 
analytically  it  will  be  useful  to  sketch  the  arguments  of  Laplace  and  those  who 
afterwards  treated  the  subject.  Laplace,  assuming  that  the  solution  of  the 
e([uation  could  be  ex])ressed  by  means  of  a  Taylor's  series,  substituted  such  a 
series  with  undetermined  coefHcients  in  the  ditierential  equation,  and  was  able 
to  determint;  all  the  coefficients  of  the  series  in  terms  of  one  of  them,  which 
remained  arbitrary.  He  had  previously  argued  that  it  was  not  necessary  to 
obtain  the  general  solution  of  the  ecpration,  since,  as  he  atHrmed,  the  arbitrary 
coustsiuts  would  be  tletermined  by  the  initial  comlitions  of  the  water  and  would 
introduce  ettects  dependent  on  this  initial  condition,  which  etfe>'ts  ought  to  be 

♦  Article,  "  Tides  uud  Waves,"  Eneyoloi.    lia  Metropolitaiui. 

+  "  Tidal  Uesearthes,"  Api>eudix  to  ITni.ed  ;^tutes  Ciiast  aud  Geodetic  Survey  Ki'ijort  for  IH',4. 

X  Article,  "  Tides."  Eucyolopedia  Britauuica. 

§  Basset,  Vol.  II.  1>.  218. 

II  No.  84a,  Siuithsouiaii  MisctUaueous  Coutribiitions. 


w 


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^ 


98 


LINO.      ON  THE  SOLUTION  OF  A  CEUIAIN  DIFFEUENTIAL  EQUATION 


(lisregavdod,  Hiiico  iu  tlie  case  of  the  sea  they  would  Vm^  ago  liavo  l)i)i'ii  over- 
come by  friction.     Considoriug,  then,  that  any  particuhir  integral   wus  HutH- 
d<>nt,  h(-  proi'oedod  to  choose  the  most  Hatisfaotory  value   of  the  eoerticiciit. 
Mis  metliod  of  deciding  tiie  proper  value  of  the  coetHcient    will    lu'   given   in 
Section  III.     It  enabled  him  to  satisfy  himself  that  a  comparatively  few  terms 
of  his  series  would  give  the  result  with  a  very  small  error.     From  the  form  in 
which  th(!  result  is  set  forth    in   tint  M«'cani(pie  Celeste  it  apjjears,  however, 
that  he  made  the  assumption  that  the  ratio  of  any  coi>thcient  in   his  series  to 
the  preceding  one  becomes  ultimately  smaller  than   any  assignable  cpnintity. 
Moreover  Laplace's  argument  regarding  the  sulliciency  of  any  particular  solu- 
tion (lid  not  occur  immediately  in  connection  with  tiie  treatment  of  ilii;,  par- 
ticular case,  and  it  therefore  appeanul  that  he  ottered  no  justitication  for  the 
assumption.     Airy   objected   to  the   assumption   on   the  ground   thr.t  it  was 
unnecessary  and  unduly  specialized  the  solution.     He  added  that,  if  the  sea 
were  boun<led  by  a  paridlel  of  latitude  instead  of  covering  the  whole  earth, 
then   the   arl)itrary   constant    could   be   determined   from    the    corresponding 
boundary  condition.     Ferrel  agreed  with  Airy,  and  regarding  the  constant  as 
being  entirely  at  his  dispo.sal,  b,.sed  his  calculations   on   the  series  resulting 
from  assigning  to  it  the  value  zero.     Kelvlti  in   his  replv  to  these  arguments 
quoted  Lai)liice's  reasoning  regarding  the  sufficiency  of  a  particular  solut 
but  pointed  out  that  this  reasoning  was  not  correcit_J2i:ucciHWH5*riffrcoiitended 
that,  as  denii.nded  by  Airy,  there  was  a  certain  physical  condition  to  be  satis- 
tied,  and  tiiat  this  condition  was  sufficient  to  justify  Laplace's  result.     He  also 
pointed  out  that  tiio  general  solution  of  the  equation  should  contain  two  arbi- 
trary constants,  and  that  a  further  boundary  condition  would  b(>  necessary  for 
till'  determination  of  the  second  of  these  constants.      He  siiowed  that,  since  the 
oscillations  of  the  water  which  are  taken  account  of  in  the  ditt'ereutial  equation 
have  a  perfe.-tly  definite  period  depending  on  the  period  of  the   disturbing 
bodv,  the  original  state  of  motion  could  not  be  taken  account  of  in  the  solu- 
tion, for,  except  for  sjjccial  dc[)ths  of  water,  the   period  of  the  latter  oscilla- 
tions would  be  ditt'ereut  from  that  of  the  formei-.     Airy  and  Ferrel,  however, 
did  not  iulmit  the  force  of  the  reasoning  by  means  of   which  Kelvin  justified 
Laplace's  result,  and  Ferrel's  later  papers  are  devoted  to  an  attempt  to  show 
that  the  determination  of  the  value  of  the  constant  is  unnecessary.     While  it 
would  seem  that  the  constant  was  correctly  determined  by  Laplace,  it  appears 
to  the  author  that  the  analytical  proof  of  this  fact  indicated  by  Lord  Kelvin, 
and  given  in  greater  detail  by  Prof.  Darwin,  is  not  complete.     It  seems,  too,  to 
be  desirable  to  obtain  the  general  solution  of  the  differential  equation,  and  to 
follow  up  the   suggestions  of  Lord  Kelvin  regarding   the  application  of  this 
solution  to  the  more  general  case  and  some  special  cases. 


■■', 


' 


"T 

T 


^ivwpqw^^Mf^^wiiqqpqQiwmnniB'ViiP^g^ 


^mmmmmmmm 


WHICH   I  /lESKN  TS  ITSIXF  IN  LAPLACE's  KINETIC  THEOIIY  OF  TIDEH.  0!) 

TTI. 

Lai'i,ace'h  Som'tion. 

0.  (n'liii'dl  (infliihi.  Apart  from  phyHiciil  coiisidoratioiis,  tlic  iirf^nniontH 
inado  l\v  Laplacu  h'om  aiialysiH  <lo  not  appear  to  ottor  a  very  ^ood  rcaisoii  for 
liis  (tvaliiatioii  of  tlic  coi'tHciciit.  In  order  to  hIiow  tliis,  it  will  Ix-  iit'ccssary 
to  ^,Mvo  Lajjlacc's  disi-ussioii  as  it  appeared  in  iiis  orij^ina!  nieiiioir.  The  form 
of  the  ecpiatioii  as  treated  by  Darwin  differs  slif^iitly  from  that  in  whi'ii  Laphien 
used  it,  hut  no  essential  ditVerence  is  introduced  hy  the  chant^e,  and  tlw^  same 
dilHcrulties  arise  in  Itoth  cases.  As  Darwin's  form  of  the  e(|uation  is  doubtless 
that  in  wliioli  the  wiuatiou  will  henceforth  bo  studied,  it  has  beeu  adopted 
here.     The  ((piution  may  then  l)e  written 


x'(\ 


(8  —  ±,r  -  ,^,,.')  „  -f  A;i,y' 


(». 


(1) 


It  is  to  be  noted  that  .r.  is  th(>  sine  c)f  the  polar  distjinci!  of  a  |>arti(;le  of  water, 
'/  the  dillerenc(^  between  the  tide  hei^dit  in  the  dynjimical  theory  and  the  tide 
hesf^ht  in  the  ei|uilibrinni  theory.  Assuming  as  the  solution  of  (i)  a  Taylor's 
series  containinjj;  only  even  |)o\vers  and  witli  undetermined  ooetWcients,  Laplace 
found  tiiat  the  coetticiont  of  a;'  remaini>d  undetermined.  He  next  proceeded 
as  follows  :  He  assumed  as  an  inte^'ral  a  sum  of  ii  (inite  nundicr  of  power.*, 
and  found  that  by  addin<^  a  certiiin  term  to  the  left  member  of  (1)  this  etpiation 
could  be  moditi(Ml  so  as  to  have  as  a  solution  the  assumed  function.  By  study- 
ing' tlu!  ell'ect  of  inoreasint,'  the  number  of  terms  in  this  function,  he  came  to 
the  conclusion  that  a  very  snndl  error  would  be  made  in  assuming  as  a  solu- 
tion, such  a  function  .vitli  a  large  number  of  terms. 

(■».   Apjiliciifl'in.     To  apply  this  treatment  to  the  ccjuation  (1),  assume 


that 


u  =A^  +  (.-l, 


E)jr+%A^r'^. 


(2) 


If  this  function  satisfies  equation  (I),  the  following  relations  must  be  satisHetl 
l)y  the  coelKeients  : 

(a) 

(0) 


.1, 


.1,=  A. 

10. 1,  -  10.L  +  ,'i^^  =  0, 

2.1,+,  \:2{k  -  D-  +  C  (/;  ~  1)]  -  2.1,1 2  {I-  -  1)-  +  3(X'  -  1)]  +  ,iA,_ ,  =  0  , 

{k  =r  3,  4,  o,  .  .  .  ,  r.) 

-2.l,^,(2r  +  3>0  +  /^'lr  =  O, 
+  ,U,+,  =0. 


(i) 
(g) 


t 


1(10 


I.INd.      (IN    Tin:  HOLUTION  or  A  <  r.UTAIN  DIFI'KUKN'I'IAI,  KglJATKtN 


Tlioro  iiro,  rtiutio  (c)  iw  iiii  idtMitity,  /•  ■\-  'i  liuotir  «i|iiiiti()iiM  to  l)o  HutiHlind  by 
?•  -j-  2  iiiikiiown  <|iiiiiilitii\s.  Il  is  oiisy  to  hod  timt  all  ciimiot  lio  Hiitisfi'id  ;  for, 
Htiii'tiii^  fi'diii  (|^j  iiikI  \v(>rkiii<^'  l)iu;k,  tlitiro  ritHult  /oro  vhIiioh  for  nil  lUv  A  s  .uid 
this  (lisiii^rcds  with  (li).  If,  however,  oiio  of  tln!  (uiuiitioiiH  Im  rrji'ctcd,  tlui 
rciimiinng  ?'  -j-  2  it'latioiis  an*  Hunicifiit  to  dcituniiiin' tlic  vidiirs  of  Ww  A'h, 
Sii|i|)os(i  (ix)  to  !)(•!  rcjt'cttMl.  Followiiij^'  riapliicf's  imihod  Irt  the  followiiif^ 
alihroviatious  I  to  iiiadu  : 

/I  =--  i,i , 
/i,  =  2r'  +  3/- , 
/i,,,  =  2(/'  -  1)-  +  :J(/'   -  1)   ~  [(/•  ^-  IJ-  +  3(r  -  1)1  /////, , 

/I...,  =  2(/-  -  kf  +  3(/-  -  /,■)  -  [(r  -  /)^  +  n{7-  -  /•)]/'//'■-*+. . 

Then  it  follows  that 


wliciioo 


yl,.+,  - 

;:■'" 

/!,,== 

vl,._,  = 

■"    A    .. 

.     '■  1  —        ''  ' 


II.' 


A', 


//,//., 
«' 


I,"'.' 
.'I 


,n,it.,it., 


-1,4, 


/«' 


/^l/^a,":. 


/':. 


Thoso  valims  of  .1 1,  -L,  .l.j, .  .  .  ,  ^l,.+,  satisfy  the  oqimtious  (a),  (li),  (c),  (d), 
((!)  (f),  l)\it  (Miuatioii  l<^)  is  not  satisfird,  and  the  fmictioii  (2)  is  th<?n'for(!  not  a 
solution  of  (1).  If,  however,  to  the  left  luoiuher  of  (1)  lie  added  the  (juautity 
—  ,'iA,..^.iX-'"^''',  eqiiatiou  (g)  becomes  the  identity 


/! 


WlllCir  I'llEHKNTH  ITHKM'  IN  I,AI'I,ArE'H  KINKTIC  TIIKOUY  OK  IIUKH. 


101 


HO  tliiit  clio  t'xpreHHioii  (2)  in  ii  solution  of  oquiitiou  (1)  tliUH  iiindituHl.  Tlio 
DOW  e(|aatinii  ciiii  Im  \vritt«!ti 

^  (1  -  •^)  ''!".  -  X '{"  -  n  (8  -  2a^  -  ,^x*)    |    Kir"  -         ^"''       ,/•-'+"      0.  (3) 
i/.ir  (f.n  '  fi^fi.^ ,  .  .  /I,.  ^  ' 

TjiipliKM*  thou  iirmicd  tliat  if  tin-  (iorri'ctivo  tniiii  wcm'o  vory  Hiniill,  only  a 
siniill  orroi  would  In-  iniidd  if  (1)  wnro  nipliUMnl  hy  (I{).  His  tlicui  procefuk-d  to 
sliow  tiiiit  iiy  tiikiii;^  /'  iiii'^c  (Miouf^li  tlio  corrective  term  could  lie  iniidc  to 
decreiise  ilidelinitely. 

7.  <  >l>}i  i-tiiins  Id  iiiitliiiil.  In  order  tlmt  tlie  diHcUHHion  jiiHt  ^iveii  imiy 
justify  tiin  clioice  of  tlie  vidiii  of  A.,,  it  HJioidd  put  in  ((videnco  hoiuo  property 
poKKessed  liy  tlie  series  when  Tjiipliici^'s  ''.tlue  is  ^iven  to  .1^,  iiiid  not  posHcssed 
1. ,  it  under  other  circuiustiinces.  All  thnt  is  iittenipted  in  the  precediu}^  ]ir(tcesH 
is  to  show  that  the  error  inadt!  in  ussuiuiii^^  us  the  intej^ral  a  Unite  iiutnlior  of 
terms  lieloni^iii}^  to  the  infinite  series  can  lie  nnuh^  less  than  any  ass!<^nal>lo 
(juantity.  i<ut  this  is  true  of  any  s<!ries  which  is  an  intej^ral  of  the  (Mpnitio.' 
and  whoso  n^j^ion  of  oorivor^onco  iH  larj^o  (Uioiif^h  lo  suit  i\w  conditions  of  the 
proliloin,  and  it  will  appear  that  no  matter  what  value  ho  f^iven  to  ,1^  tho  refj;ioii 
of  (•onverf:;ence  of  the  rc^sultin^  s(!rics  is  still  larj^e  enough  to  suit  tho  |)urposo. 
Moreover,  it  is  not  definitely  proven  Unit  a  vanishinj^  correction  to  the  e(pni- 
tion  nocoHsnrily  indicatc^s  a  corresponding  vanishing  of  tint  error  due  to  tho 
assumption  of  a  finite  numl)t>r  of  tfu'uis  instead  of  an  infinite!  numlier. 

H.  liiiliitiiiii  liitirmii  i'in')'i'i'tnin  bi  ii/nitfidii  (Iid/  i'rr'>r  in  Kri'ii'n.  Tho  rela- 
tion betweon  tho  corroction  to  tho  equation  and  the  error  in  tho  value  of  tho 
dopondoiit  varialtlo  can  he  shown  j)orhaps  more  clearly  in  the  following  man- 
ner, assuming  (Certain  propei'ti(!s  (if  the  series  used  which  will  luMleduceil  lat(^r  : 
The  difVerentiai  ('((nation  (1)  can  lie  regardcid  as  a  linear  reflation  connecting 

tho  (inantities  ",   ,    ,  and    ,  „  in  which  the  coofH(Ments  ans  rational  entire  func- 
'  il.i;  il.i- 

tions  of  X.     TIh!  function  k  is  to  ho  oxpross((d  liy  means  of  an  infinite  sorios. 

This  s(!rics  will  have  a  ciu'tain  circle  of  convergence.     For  all   points  iiiit/iln 

this  circle  th(>  corresponding  series  for  and  .,  will  also  i-onvei'ge.  Sup- 
pose the  circumferonco  of  this  circle  of  convergence  lies  entirely  outside  of  tho 
houudary  of  tho  region  in  which   the  independent  variable  is  to  vary.     Then 

when  ",    ,    ,  and    ,  .,  are  each  replaced  by  tho  finite  number  of  terms  from  tho 
(l.r  (/,/;-  '  '' 

series  expressing  their  values,  certain  errors  will  bo  made  in  tho  case  of  each. 

These  errors  will  each  become  less  than  any  assignable  quantity  if  the  number 

of  terms  be  sufHcicntly  incroasod.     Let  (1)  then  Ih;  written 


-,  (hi.    , 


.j 


I  I 

i  1 


i 


S.     i 


102  l.lNd.       ON  THK  KOT.rrrON  OV  a  CKIITAIN   DlI'l'-KliKNl'IAl,  WiHATION 

wlionw/,  ,V„  ;-,  iiiwl  o  iuv  (iiiito  for  all  viiliios  to  bo   consid.'nMl.      L.'t  the  Inui 

VMJii.'s  ..f  "'  "  ,  "'"  ,  iiiul  "  wlu'ii    foun.l    from   tl'o   iiilii.  Ic  series  l)o  .1,  /A  aiul 

r';  and  1. 1  i,,  J.,  and  £,  l)i'  tlio  (•orrcHi)oiidint'  oirors  mad.'  in  taking  Cu;  linito 
niimlii-  of  terms  \,,r  each  i>f  tin'  Mut'c  (jiiantities.     'I'licn 


00 


|'|„,„  tl,,'  .•uiiv,-ti(.n  to   lieadd-d   t<.   the   left   munib.ir  of  {k)  to  make  it  an 
,i,,iljtv  is        I'/s,    I   ,;,c;    i    i'-i)-      I'nder  ;ii<!  eircnmslanci^s  assumed  al)ove,  in 


»'( 


n^feivnee,  to  tlie  re^^if.n  of  (•onv.M-fi;ence,  tliis  correetion  will  heconn!  indeliniti'ly 
Hn.all  when  Ih.'  muuiIkt  of  tl  e  terms  is  snlVieientlv  increased.  Hut  oilier  eases 
,„,iv  ocenr.  Consid.^r  the  ease  of  Ih.!  inlinile  serii^s  in  whieli  .1,  is  given  any 
„ti,cr  vahn^  than  that  assigned  by  Laplare.  it  will  afterwards  appear  that  this 
series  converges  for  all  values  of  .-•  within  th.'  unit  circle,  and  also  for  ,/•  =--    i    1. 

The  series  for  '{"  and  "',"'!  ,  liowexer,  art nv.'rgent  only  for  j.oin.s  within  the 

unit  .•ireie.  it  is  clear  then  that,  for  all  valr.es  of  ./•  less  than  unity,  £„  £„  and 
£,  can  be  made  less  tiian  any  ussignabh  .iuantily,  and  that  therefore  th.^  same 
is  true  for  «£,,    I    (V,:,,,    |    ^s... 

lint  ,'■  must  i)e  considered  for  all  values  up  to  and  inchuling  unity,  and  it 
do.s  not  appear  thid,  as  r  app  -oaches  unity,  the  correction  must  necossarily 
i„delinitel\  diminish,  siiH'c  i,  is  the  only  on.'  of  the  three  .piantities  which 
mdclinitely  diminishe:  .  .Moreovr  the  s.-ri.'s  is  a  satisfactory  solution  if  only 
c,  ..,in  b(!  made  imU'ti.dtely  small  for  .'•  .1.  On  th<'.  other  hand,  it  would  also 
a,l)pi'ar  that  the  <  orrection  migiit  i)i'  evanescent  wht^n  £,  is  not  so. 

'.».  .[s.siiiiij'ti'iii  i,iii'ir<ihiil  In  l.>i/>iiicr'.:  (iKsiniipl'oui.  The  same  set  <)f 
,., Illations  for  the  determination  of  the  (^oetUci.mts  will  'le  olilained  if  it  be 
assumed  that 


£ 


A. 


'  '     rr^    0   . 


I'or  then  it  w<iuld  !>■■  proper  to  assunn'  tiiat  tli piidions  (onneding  the  co.d'- 

licieiitH  es.uitualiy  took  the  sam(^  t'orn;  as  (f).  l-'or  all  other  se.M's  in  which  the 
,vl,ili(.ii  jiist  written  is  not  true,  tiie  e.iuaiion  (f)  is  not  satislied  when  a  finite 
number  of  terms  is  taken,  but  has  to  i)e  corrected  by  the  a.hlition  of  a  term. 
|''rom  this  j.oint  of  view  then  liajilace's  proeesH  would  seem  to  noceKSiirily  lead 
to  a  series  convergent  over  tlu!  entire  plane. 


WHICH  I'ltKsKNi's  ITSKJ.K  IN  i.ai-i.ack'h  KiNirric  THKoitv  ()!•:  'I'lDiOH.  iu;i 

TV. 

The  Somition  or  thk  DiI'Tkhkntiai,  Equation. 

Ml.   (7i(iriic(i;r  nj'f/n  i/iffi/ra/.     Ijiipliicc^'s  Holulioii  having  Ixhui  I'oiisidon'd, 
tliii  gcnoriil  iiitcgnil  of  llic,  luiiiatir:;, 


^^"{1  -  x')  '''".        ,' '';'        n  (H        2,:^  --  fix*)  ^-.       AK,^  , 


(I 


iiia_v  now  1)0  Houglit.      In  this  (■(|ii,\tion   it  nuist  hv  rcnicnilicrcd  tliat  ,f  ::     sin  I) 
wlitiro  (\  iH  a  puhir  distanftt.      It  i.s  nc^cos.nai,)'  (irst  to  >solv(^  (he  anxihary  (!(|ua- 


ti'ui. 


iPn  ,l,i 


«=(l~.;^)y       .v"l'^        ,,(H        2,r     ^V) -.:(), 


(7) 


Thr  following  g'.'Mi^ia!  tlicoifniH  will  !»!  nscful : 
'rnr.ouKM  I.   /;/  tirdcr  //iiif  t/if  ci/iKilidii 

t<hiill  /id re  II  iiiilijH'Hil,  lit  iii/ci/ra/s  of  th,   fiiriii, 
n-.*-(/^_,  log"    '.     I     /^,^.Jog"    ^,;    I-    /^,   Jog"   Iv 


ihi 
(1.1' 


r„<|^A^    (H) 


/',  i"K.''  I  /;), 


ii'ln  re  /'„  /',  /'^,  /',,  . .  .  ,  /'^^  I  are  e.rprexnihle  in,  (lie  ihiii/ihor/nxnl  of  x  ~-  0 
/«  seriex  of  potiit.ive  tiiiil  iieijai'iee  pinners  of  .r,  fhe  iniviher  of  iiei/dtive  poire.rs 
111  eiie'i  heiiKj  '•'ii'if,'.  It  is  iireessd  r;/ n  ml  siijlieient  t/mt  for  i  mli  nf  the  eueffieientH 
(i_t  the  i>iii<itii>ii,  Kiie/i  (in  /(„  the  paint  ,/■  (I  xhiill  lie  an  urdiniiri/  point,  or  ti 
pole,  irhdse  order  of  nmlti/ilieiti/  does  not  i.reeed  i. 

'l'lu^S((  integrals  will  constitute!  one  or  -v.  groups  of  the  form 

"i  --  .'•'  J/|  , 

n,  .-.,,•'■  (.IAI.,g.,;  -I    a;), 

"3  -.'••(, I/.,  log-.,;    I    v\'.,  log.,'    I     //,), 


«,=--.,••■(, I/Jog''    './■    I     ,VJog«   V'    I     ...), 

when!  .!/,,  .)/,,,  J/„  . .  .  ,  .I/j.  dill',)i-  only  l»_y  constant  fuctors. 

Thkoukm   II.    '/'he,   inte(/rdl  of  the   ('(pdition  (H)  irill   !„■   rontiinious  und 

■nioiioijenir   fo'  oil  edliies  oj'  .,•  _/;*/•  (/'/,/,•//  thi'  rmj/irienls  p^,  p„  p.^ p^  ore 

eoiiliiniods  diiil  iiioiHK/enie,  and  it  eon  poxursr  no  eritiedl  points  irhieh  are  not 


1 1 
I 

I  % 

I     % 


104  LINO.      ON  THE  HOLUTION  OF  A  CEUTAIN  niFFEUENTIAL  EQUATION 

aim  critical  points  of  om  or  more  of  the  ,u>,^i/identK     ft  may  not  hare  critical 
pohit>^  at  all  the  critical  pointH  of  the  coeffir],'M><.  „  -      ,       # 

The  proofH  of  theso  tlieoroiuH  iiitiy  l)c)  foiuiil  in  Jordan's  Vourx  <r ivnalyse. 

If,  then,  equation  (7)  be  put  in  the  form  (8)  ami  these  theorems  applied, 

it  is  clear  tiiat,  ,   ■      ,,      ,  « 

(1)  Equation  (7)  has  on.  intc^^ral  wliich  .^an  be  expressed  in  the  form  of 
a  pow(!r  series  ^■,  and  a  soeond  int..^ral  whicli  can  be  expressed  in  the  form 
^V,  +  il',  loK  X,  wliere  c'',  and  v'',  uro  expressible  as  power  series  and  V^;  -=  a  con- 
stant c  (when  c  =  0,  the  second  intej^ral  is  also  expressible  as  a  powei  series) ; 

(2)  The  general  integral  of  (7)  c^an   have  no  critical  i)oints  except  at  0, 

±  1,   ±  00  . 

i'.h  If  ./;  be  replaced  hy  -  *,  the  e(piati()n  remains  unaltered;  whence  it 
follows  tiiat  tlu!  function  has  tlio  same  character  at  x  =-  —  a  as  at  x  =  4  «, 
and,  in  paiticular,  if  the  general  integral  of  (7)  has  a  critical  point  of  any  sort 
,^t ,;.'  =  +  1  it  has  a  critical  point  of  exactly  the  same  character  at  .'■  =.  —  1. 

It  follows,  too,  that  if  U  is  any  particular  iut.^gral  of  (1),  the  complete 

integral  is  ,         s         rr 

u  =  rt,v''  +  '1,{,/k,  +  c<i'  log  ,/!)  +  ff, 

when  c  ---  0. 

11.   /)e>l action  of  the  complementary  fanction.     It  is  correct,  then,  to 

assume  as  one  iuteg'-al, 

•v  being  a  positive  integer.     Thou 

^    -.-  2,.  (hh  -f  rt<)  A.^  , 

n 

l,.{>a  +  rs)  (//.  -  1  -f  r,s)  A,,i:"'"-+'-'  , 


<l-a 
<lx' 


The  result  of  substituting  in  (7)  is 

v_,  (;,,  -f-  rs)  in,  -  -  1    I    r.v)  71,,/;"'+"  -  I',.  ("'■  +  rn)  {m 

-1-  I;2vi,y"+-+'M  i',.M,.c"+'+"  =  o, 


—  1   4    /■.v).l,,X''"+''*"'' 


(9) 


♦  Vol.  Ill,  Arts.  14«,  "J2,  ll«. 


^1 


WIIK'II   I'liEHENTH  ITHELF  IN  LAI'LACE's  KINETIC  TIIEOUY  OF  TIDES. 


105 


identioilly.     Tlio  cocfliciont  of  x'",  tlio  lowest  powttr  of  x,  must  vaiiiHli  ;  wlienco, 


wince  A„      0, 


111' 


'ini,  —  8  =  0,  aud  iii  =  i  , 


Tlu!  valuo  of  .V  iimst  be  2,  for  if  .s'      2  the  coofticiont  of  3-'"'+-(loe8  not  vanish. 
fhv  result  of  snhstitutini,'  .y  =^  2,  in  =  4  in  (0)  is 

i',  (4  +  2/0  (3  +  27')  A,ar'+'    -  i,  (4  +  2r)  (3  +  2/')  .l,,.y-'-+" 

0  (I 

—  i',  (4  H-  2r)  A  ,ar'-+i  -  i',  8/1  ,,-f-'+' 
(I  II 

+  i',.  2yi,„Tr'+"  +  i;  /9yi,xc='+"  =  c .  (10) 

From  CIO)  the  e(iniitions  for  the  doterniiuation  of  tlic  coeffieionts  are, 

(11) 


1(>J,  -    10vl„=--  0, 
2^  (2^-  +  G)  A  ,  -  'J/'  (2/('    I    3)  .1 , , ,  i^  ,Jvl,_,  =--  0  , 
(/!•  =  2,  3,  4,  5,  .  .  .) . 
Tlius  each  cootlicieat  is  a  uiuitiph)  of  .1,,  and  one  integral  (;an  l)e  written 

yip  -i|,  ^i|, 

=  a.''  +  (y  H  *' V  +  r:,«"'  +  . . . . 


(12) 


(13) 


It  is  aasy  to  vtuify  that  A',  */,  is  the  series  written  (h)wii  by  Airy  in  his 
article  in  tli(!  I'liilosoplucal  I\riij,'a/.in((  as  the  correetion  to  Laplace's  series,  J\\ 
being  an  arbitrary  constant.  On  applying,'  to  the  ecpiation  the  tlieory  set  forth 
by  HetVter,*  it  ap|)ears  that  there  is  no  series  corresponding  to  tlie  root 
III  =  -    2.     Till!  second  integral  is  then  obtained  in  tlu^  form 


I'loni  this 


"•2  =  ^''2    I    v''i  loK  •'•  ==  ^'-1    f-  '1'  "i  log  X  . 
iIk.. '/v''i    ,     i,  1 (III,    ,    A' 


dx\       (Ix  "      (tx        X 


iP 


i-i  _  ""v 


da?        ih. 


J  + 


2.1' <A, 


.1' 


+  A'Xo^x 


</-ii^ 
ilx- 


'  Liuc'irf  Dill'ereutiikl  gliMchuup'ti,  §  Hi. 


?1   V 


106  I.IN-0.      ON  THE  SOLUTION  OF  A  CEUTAIN  DIFFEllENTUL  EQUATION 

These  values  being  substitnte.l  in  (7),  there  results 


a? 


_JV,  (2-.r=)  +  ^l'-2,/!(l-.'-)'^^ 


..  ,J 


+  A-Xo'^x    ,j;-(l  -  .'.  )  ^/,,.  ,/,,,  '>  '       J 

identicallv.     If  th.  substitution  ,',  ==  ^V.  '-'  '"-^^  -^^  '^  ^^  ^-"'^  "'  '"^"^^ 
that  «,  is  a  soUitiou  of  (7),  (15)  roduces  to 


..a--)';^-'--';.;^-"^'^^'-'"'^''" 


) 


-  (2  -^  a^)  ",  '!-  2*  (1  -  a^)  ^^  =  0  . 


(1(5) 


It'  now  it  be  assunietl  that 


and  th 

is,  as  before, 


^'v,  =  2', /A ■'•■"+".  ^^''^ 

.  substitution  bo  nnule  in  (IC),  the  equation  for  the  determination  of  m 
}  _  2»/«   —8  =  0,  whence  m  =  4,  —  2  . 


VI' 


The  theory  shows  that  //*    -  —  '^  is 
«  =  2.     The  substitution  in  (IG)  gives 


2  is  the  root  to  bo  taken,  and  that,  as  before, 


iV  (2r  -  2)  (2r  -  3)  //..r'^^  -  %■  (2^  -  2)  (2r  -  3)  T/,.^^'- 
-  1,  (2r  ^-  2)  B,^:^-'  -  X  «  /A-r^  -^  +  f  2 /A-  ■'-'• 

+   V   4  (;•  +  2)  (\..P-^'  -  iV  4  (>■  +  2)  0^'+"  =  0  , 
in  which  the  r's  are  the  coefficients  in  the  series  «,. 


(18) 


'       LI 


WHICH  PREHENT8  ITSELF  IN  LAPLACE's  KINETIC  THEOKY  OF  TIDES.  107 

From  (ID)  are  derived  the  relations 

0  =_-  8/A  +  4Z/, , 
0  =  8/;,  -  2//, 
6  = 


?Ji., 


(19) 


7  -  lor;  =  iG/;,  -  10//,  +  ^n„_ , 

{U  -  5)  (\_,  -  {ik  -  2)  (\_,  =  {2k  -  4)  {^k  +  2)  //,+,  -  (2^'  -  4)  (2/J-  -  1)11, 

(k  =  4,  5,  G,  .  .  .) 

These  equations  determine  all  the  coefficients  in  terms  of  li^,  the  coefficient  of 
x\  wiiich  remains  arbitrary. 

Since  u^  is  a  solution  of  (7),  it  is  wear  that,  if  if  is  a  solution  of  (IG),  so 
also  is  f  +  yi',?<|,  A'.^  being  any  constant.  Moreover  ;/,  starts  with  the  fourth 
power  of  X.  Then  if  any  value  be  assigned  to  //,  and  the  resulting  value  of 
^''3  be  denote.l  by  <f  we  can  write 

^  is  a  series  of  ascending  entire  powers  starting  from  B^x-'-  in  which  every 
coefficient  is  known.  />'.,  may,  if  it  is  desired,  be  taken  to  be  zero.  The  com- 
plete complementary  function  of  (17)  then  is 

''  =  ^I'l  (v''2  +  'I'x  log  x)  +  A'.^i, 
=  ^\  {'/•2  +  ^l'"!  log  x)  +  A'^%, 

=  ^I'^-l'l  (V'':.  +    "l  log  X)   +   A\,H, 

=  A  ((f  A-  n^  log  x)  +  Ba,  .  (20) 

12.  The  jmrticuhir  integral.  It  remains  to  determine  a  particular  inte- 
gral of  the  complete  equation  (1).  The  character  of  </,  and  of  the  absolute 
term  of  (1)  makes  clear  the  existence  of  a  particular  integral  expressible  in  the 
form  of  a  series  of  positive  entire  powers.     Assume  then 

U  .-  A,  -h  (^,  -  K)  x^  +   i'  A,.x-'- . 

After  substitution  in  (1)  there  result  the  relations 

^„  =  0  . 


2.4,+,  r2  {k  -  1)^  +  G  {k  -  1)}  -  2.1,  {2  {k  -  If  +  3  {k  -  1)}  +  ,?.4,_ ,  =  0  . 

{k  -  2,  3,  4,  .  . .) 


■H  t 


;if  ' 


111  1^ 


108 


UNO. 


ON  THE  SOLUTION  OF  A  CF.ItTAIN  DIlKKIiKNTIAI.  K(H'ATION 


All  the  coofficiouts  ure  «iven  in  terms  of  h'  mul  .1,,  the  latt.a'  be.n^  arb.trury. 
Since  ..nlv  a  partic-nlar  integral  m  re<,uirecl  any  vah.e  n.ay  be  given  to  A  In 
,„.oeee.lin\r  to  a  choiee  of  a  value  it  is  interesting  to  follow  the  n.etl.o.  by 
'vhieh  Laplace's  continuecl  fraction  is  obtained.  From  the  relations  ^vntten 
just  above,  it  can  easily  be  deduced  that 


■A ,+,  __ 
A,   "2(2/'-^  -I  H/') 


2(2/'- -h  ()/O^UW '•*■+' 


(/•  =:  2,  3,  4, . . .) 


A..                   fi 

'■      2;~   2(2.r-:   3.1)    - 

(2.1^-!   G.1),V 
2(2.2-  ■:   3.2)- 

|2 

2  12(//  -1)'  +  '^(" 

-1)1-  2|(2//^   1- 

and  A  ,     =  /'• 

The  MSSHnii)tion, 

J^    -^1,1+1 

(2.2--.  n.2),i 
2  (2  .  3-  H-  3  .  3)  —  ■  ■  ■ 

2(,,  _iy  4- fi  ("-!)} /^ 

3//)  -(2//M-  <>")]  A„+.,/A„+, 


(21) 


nives  th"  value  of  A.,  in  the  form  of  an  infinite  continued  fraction.     It  is  per- 
missible, if  convenient,  since  any  value  of  A,  may  be  taken.     It  was  made  by 
Laplace  in   the  Mc^-caninue  C.'^este  apparently  without  justihcation  ;  but,  as 
has  been  seen,  Laplace  believed  in  the  sufficiency  of  a  particular  solution  and 
considered  the  resulting  series  as  a  satisfactory  solution  without  the  addition 
of  the  complementary  function.     The  assumption  (21)  so  affects  the  coeth- 
cients  that  th.  .cries  converges  for  all  finite  values  of  .r.     It  is  not  necessary 
that  the  parti.-nlar  integral  shoul.l  converge  for  points  outsi.le  the  unit  circle. 
It  is  convenient  from  a  mathematical  point  of  view,  however,  to  choose  this 
series  as  the  particular  integral,  for,  if  it  were  necessary  to  study    he  function 
for  points  outside  th  .  unit  <-ircle,  it  would  be  sufficient  to  obtain  the  cx,mple- 
uu.ntarv  function  in  the  form  of  a  Laurent's  series  while  the  series  just  found 
would  ;.n-ve  again  as  a  particular  integral.     The  assumption  (21)  i.  seen  to  be 
ecpiivalent  to  that  involved  in  Laplace's  original  proce.ss.     From  the  reasoning 
of  this  section  it  is  clear,  too,  that  when 


£ 


A., 


then 


Xi    A„ 


1. 


it 


WHICH  I'1!i;si;nts  ri'sEM'  i\  r,Ai'LACK.s  kinetic  theouy  of  tiueh. 


10".) 


From  tliis  ))oint  of  vi(!\v,  too,  it  is  (iloar  tliut  tlio  sorids  under  coiisidfriitioii 
<  oiiverges  ut  ItuiHt  for  (tvory  jjoint  wilhiii  tlio  unit  circlo,  mid  tlmt  if  it  convcrgeH 
for  II  groiitor  circlo  of  coiivergenco  it  coiiverj^cH  for  overy  fiiiito  viduo  of  u-. 
For  tlio  Kfiko  of  defiiiitiMicss  Liiplaeo'.s  viihus  of  A.^  will  ho  donottul  by  /-  and 
tilt!  serieH  which  furniHiios  the  pjirticular  iiit<'<5ral  of  (1)  will   ho  denoted  hy  V. 

I'ii.  I'mpcrtien  of  the  t'ompleU'.  iiil''(ir(il.  The  (•oini)lete  integral  of  the 
equation  (1)  ean  then  ho  expressed  for  i)oiiits  within  the  domain  of  the  origin 

i»y 

u  =  A  (if  +  u,  log  ./•)  ^f  11  „,  4   V  .  (22) 

(1)  Convergence : 

The  most  general  integral  in  the  form  of  a  positive  power  scsries  ean  lie 
writt(in 

a  =  B  */.,  4-  V  . 

The  relaticiiis  among  the  ('oetHcieiits  of  such  a  series  sliow  that,  unless  II 
is  zero,  the  circle  of  convergence  is  of  unit  radius  ;  and  when  //  is  ztno,  the 
circle!  of  eonviu'gence  has  an  indefinitely  great  radius.*  It  follows,  then,  that 
^/|  converges  only  for  points  within  or  on  the  circh;  of  unit  radius.  Again, 
y''.,  =  ^  +  A. ,11^  does  not  coiivesrge  for  points  outside  the  unit  circle.  Then  if 
converges  for  points  within  the  unit  ci'cle.  It  is  conceivaljle  that  //j  may  have 
been  chosen  so  that  ^  shall  converge  all  over  the  finite  part  of  the  plane. 

(2)  If  A  --  li  ^=  0,  the  function  has  no  critical  point  exce])t  at  infinity. 
If  A  .^  0,  />       0,  the  function  has  critical  points  at  ±  1,  ±:  x  .     If  II  —  0, 

A  -^  0,  the  function  has  a  critical  point  at  0,  ±  co  ,  and  (except  for  one  ]iar- 

ticular  choice!  of  //,)  at  ^t  1.  In  addition  to  the  singularity  of  ^'  at  ./■  =;  0 
integrals  of  this  class  and  integrals  of  the  general  class  have  a  singularity  at 
./•  =  0  due  to  the  singularity  of  log  .'■  at  that  point,  and  are,  in  addition,  many 
valued  at  any  point  owing  to  the  properties  of  log  x.  This  iiideterminateness 
will  he  removed  if  it  is  assum(!d  that  for  positive  real  valr.es  of  .«  the  result 
shall  be  real.     Again,  when  //  =~  0, 


But 


ilx 


1  r^/tf  ,  ,      i/ii 

A  I  ~J-    -t    log  j:     , 


L 


'Ix 


h; 


-r 


1      1 


£[; 


It) 


,1,1    — 

~j-  I    1  —  x^ , 


J   9= 


^ 


dx 


-i   |(log,c)  I    1 


dx 


*  The  (Ictiiiloil  proof  of  this  fnot  is  (jHoted  in  Section  VI. 


Y     it  j 


.  5 

«1 


110  LINO.      ON  THE  SOLUTION  OF  A  CEHTAIN  DIFFEUENTIAL  EQUATION 

It  will  afterwards  be  sliown  that  for  the  complete  integral  (22) 


a  tinite  quantity, 


aud  that 


J  »=»/» 


f^^O  =  a  finite  quantity. 


.•.  for  all  integrals  of  the  form 

n  ^  A  (f  -H  "i  log  ./•)  +  V  , 

[  "'"  1  =  a  auito  quantity  or  zero. 

[,I0  '■  ^ 


I  9^ir/2 


V. 


The  Determination  of  the  Constants  foh  Laplace's  Case. 

14  Tin-  pinjdml  coiuVdhn,.  It  having  been  agreed  that  the  constants 
shall  be  determined  to  suit  the  boundary  conditions,  the  case  discussed  by 
Laplace,  where  the  whole  earth  is  covered  with  water,  may  new  be  treated. 
The  expression  for  v,  as  given  in  (22),  has  an  infinite  value  when  x  =  0,  unless 
A  =  0.  Since  there  cannot  be  a  tide  of  infinite  depth  at  the  pole  it  is  neces- 
sary  to  make  A  =  0.     The  remaining  expression  is 

n  =  BUi  +  I'. 

Airy  and  Ferrel  contended  that  this  was  the  exact  expression  for  u,  and  that 
JJ  could  be  giv  n  any  value.     Ferrel  determined  it  by  the  condition 

li+L^O.  (23) 

Lord  Kelvin  poiiUed  out  that  owing  to  the  symmetry  of  the  disturbance  in  the 
two  hemispheres  the  meridional  displacement  of  water  should  vanish  at  the 
equator.  The  expression  for  the  meridional  displacement  is  the  product  of 
two  terms  of  which  one  involves  the  latitude  and  the  other  does  not.  The 
factor  involving  the  latitude  is, 

(24) 


s  = 


i)ii  ain^O 


■^«  +  2./cotw| 


When  II  =  r/2, 


Am.  i</tij  g,„/i 
*  0  =  co-latitude  or  polar  distance. 


,^ 


WHICH  I-nEHENTS  ITSELF  IN  LAPLACE  8  KINETIC  THEORY  OF  TII>EH. 

at  the  equator  it  is  noeossary  tliat 

(tti 


dd 


=  0, 


111 


(25) 


But 


•.  it  is  necessary  that, 


mi 


-. ,  fhi 

I    1  —  x'  -,    . 


£1'''-</:h« 


(20) 


15.  Proof  that  li  =  0.  In  order  to  prove  that  tlic  ecudition  (2G)  requires 
tliat  B  shall  be  zero,  the  function  will  be  considered  in  the  neighborhood  of 
X  =  1. 

Assume 

x-.-=\^y;  yil) 

then 

(lu (lu       (ihi d-ii  _ 

Tfx       ihj '     <l,ii'       dy-  ' 


and  equation  (7)  takes  the  form 

,   (I'-u 


dii 


(22/  +  !Sf  +  4y'  +  2/')  ^t^  +  (1  +  V)  J^^ 

+  n  { {U  -  ,5)  -  4  (1  +  -i)  7/  -  2  (1  +  3, J)  f  ~  i,ii/  -  ,i//'  |  =  0  .      (28) 
Theorems  I  and  II  apply  to  (28)  also.     Then  assume 

it 

.V  being  a  positive  integer.     The  equation  for  the  determination  of  i/i  is 

2m-  —  wi  =  0  ;  (20) 

whence  m  —  0  <jr  -|-  ^.  Also  .v  ^  1.  The  two  integrals  are  expressible  in 
series  form.  Moreover  the  relations  connecting  the  ^'s  are,  for  Ijuth  values 
of  911,  such  that  each  coefficient  is  given  as  a  multiple  of  the  first  one.  The 
two  independent  integrals  of  (28)  may  then  be  written 


Z/i 


1  -I-  1\.  o.,j/ 


(30) 


Vi  =  y*  +■  -,■  ily'"^^ : 


the  «'s  and  ,9's  being  known. 


i| 


Ml 


(:u) 


112  I.INd.      ON   I'HK  HOLimON  OF  A  CKUTAIN  Uin'KltKN  I'lAl.  KlH'-VTION 

The  coiupleto  iiitogml  of  (28),  tluni,  is 

n  =  6-,y,  -\-  t'iy-i . 

If  now  tho  Hubstitutiou  (27)  bo  lumlo  in  (1),  an  o(,uiition  results  which  .lilVoM 
from  (2S)  only  by  tho  ox,.rossiou  -  K,^  (I  !  yf  i..  tho  right  h.iml  uuMuber. 
Of  tliiH  o.inution,  tho  eomplomontai-y  function  in  givou  by  (:U).  To  con.p.ete 
its  intcn-ution,  it  Ih  necessary  to  tiu.l  a  particular  intogml.  The  character  of 
y  .ui.rof  th.'  absolute  term  -  h]i  (1  +  l/f  make  it  clear  thai  a  particular 
i.ltegral  can  be  obtaine.l  in  tin*  form  of  a  series  of  positive  entire  povyers. 
Let  this  intc-ral  b.^  .lenot.-.l  bv  ):  It  is  not  of  iniportaiun*  that  its  coethcionts 
be  calculate.1.  TIum.,  in  the  neighborhood  of  .-•  =  1,  or  ..f  >J  -  0,  tho  integral 
of  (28)  cau  bo  exprosseil  in  the  form 


^\'J\   + 


Then 


Also 


(la 


(IVl  ^  ,,,  "yi  .,.  " 


dy., 

<iy 


dV 

dy   • 


where  /  =  I    —  1  • 

da        dii       .. — 
•■•      dl)  ^  dx  '   ^  ' 


dy      ' '  <ly 

■■=  I  -(z/-  +  2//)  =  ;2/4r2'n/, 


:-?  ^  .,/^4  (2  +  z/)5  j!j  +  'vy^  ri  +  y)'  % 


+  'y(2  +  y)^' 


■'y 


(32) 


When  //  -  0,  the  tirst  and  thinl  terms  on  the  right  vanish,  and  the  miiiaio 
term  becomes  . 

{du/dO)e.nyi  -  6v7i   2  .  (33) 

In  order,  then,  that  at  th.>  e.piator  dn/dO  -  0,  tho  function  must  be  such 
tliat  <■,  -  0.  15ut  if  '■,  -^  0,  tho  point  ^'  =  1  is  an  ordinary  point  of  tho  func- 
tion, while  if  <-,  0  the  point  a;  =  1  is  a  branch  point  of  the  function.  If  the 
point  .'•  =  1  is  an  ordinarv  point  of  the  function,  so  also  is  x  =  --  1. 

In  this  case  th.>  fun.-tio.  has  no  crili.^d  point  in  tho  finite  part  of  the 
plane,  and,  if  expressible  in  ihe  neigliborhood  of  tho  origin  .'•  =  0  by  means 
of  a  Taylor's  Hories,  that  series  will  converge  for  all  finite  values  of  the  varia- 
ble j:.  If  the  point  .'•  --  1  is  a  branch  point  ..f  the  function,  and  if  the  function 
is  expressible  in  the  neighl)orhoo.l  of  the  origin  ./•  =  0  by  moans  of  a  Taylor's 
series,  that  series  will  have  a  circle  of  convergence  of  unit  radius.     It  follows. 


rt 


WIIICII  I'llKHKNTH  ITHKI.K  IN  I.AI'LACK'h  KINETIC  TIIF.dllY  OV  IIDK.S.  1  Ut 

tiicii,  wli,  I  />'  {)  iuid  u  V,  tlmt  (..  =  0  and  (*/«  '/")»=it/'j  --  ".  i>ial  when 
//      <t,  tlmt',,      (»  anil  ('/«/'/")«  T /a      0. 

'I'liiis  it  iH  Hccii  that  tlio  cDuditioii  stilted  by  Lord  Kelvin  roiinin's  that 
//  =  0  and  II  ---■  V.  ConHoqnontly  the  series  us  written  l»y  Laplaee  is  the 
complete  solution  for  the  eas(!  of  an  earth  completely  covered  to  u  constant 
depth  hy  water. 

Note. — Eipiation  (Jj;})  {^ives  an  inniginary  value  iox  {ilii/tl»)„,  ^^.^  when  thi' 
arbitrary  constant  i\.  is  taken  real.  It  is  evident,  however,  that,  when  real 
values  of  the  function  between  .r,  ■-  U  and  x  —  1  are  desired,  i:^  must  he  taken 
purely  imaginary  ;  for  it  was  assumed  that 


1  +  y, 


(27) 


KG  that  when  x  is  h>ss  tlnin   unity  //  is  ne>^ative,  and  y*  is  a  pure  imaj^imiry. 
The  <:.)/■  will  he  real  when  Co  is  purely  imaginary,  and  it  follows  that  then  //.,  is 


also  veal. 


VI. 


DaUWIN's  PllESENTATION   OF  Loltl)   KlXVIX's   PitOOl'  THAT   li  MIST  JiK  ZEIt(J  WllKN 

1().  //(ti  ii'!/t\s  iiiyiniii'tif.  The  function  u  -~  />»/,  i  V  may  lie  regarded 
as  u  single  series  of  even  positive  integral  powers  commencing  with  the  fourth 
and  having  tlii^  coetlicient  of  .'•'  arbitrary.  Tt  lias  already  been  seen  what  rela- 
tions eonniH-t  the  ct)cllicients  and  detine  them  in  terms  of  tlie  coelHeient  of  ./•' 
{A.2  say).     It  is  known,  too,  that  when 


£ 


.  -''•ii+i 


=  0 


then  A,  =  L. 

Suppose  now  that 


£ 


*  A 


A 


"+'       0, 


but  =  «""',  a  finite  quantity.     Then 
A„+,_2n  +  3 


^l..+i 


/i  A" 

0       2n{2)i   +  6)  J„+, 


2n  +  3 I'i 

2ii  +  G       '2«  (2)1  -f-  ()) 


T-,  («  -  /') 


(34) 


IPUPVHP 


Mi 


114 


t.INd.      OS   rilK  SOMITION  OK  A  (  T.UTAIN  DIKKEItENTrAI.  EQUATION 


' 


whero  /,  tenuis  to  /.to  wIum.  >,  »..-comcH  iiuletinitoly  Kivat.     D.viwii.'s  ai^iii.ionf 
XH  alon^  thu  followiiin  linuH  :  NVlifii 


£ 


■  ^»+'  >.  0 


tlicn,  for  liii^'u  vahu'rt  of  /^ 


,.^,a!-^'+'^  ~  2«.  +  «}  •'         ['       2(«. +  3)J  ^         2// J 


ar 


U''ii>'ly-  ,.       ,  ,, 

lint  if  (1  -  .'•')*  Ijo  oxpamlod  by  tlio  Miioiiiinl  tlu^orom  the  ratio  of  tlio 

(«  +  l)th  term  to  the  >dh  term  Ih  [  1  -  ,j*^^  j  ar.     Conse.iuontly,  in  thin  cuho, 

it  irt  i)OBsil)le  to  write 

'  „  =  .1,  +   //,  (l  -x")», 

where  yl,  aiul  A',  are  finite  for  all  values  of  ,'■.     A  similar  argument  beinfj;  made 
in  the  case  of  the  Heries  for  iiii/</x  it  follows  that 

(iu/(lx  =  ('  -\    /Hi  - -^rri  , 

where  ( '  and  />  are  finite  (and  not  zero)  for  all  values  of  x. 
15«t  .,  ,        ,. 

t/„/,/ii  -  ii,i/</j!  (1  -  jr)'-  -  ^/(i  -  *-)'  H  Jf; 

.:     {dn/dfl)e=„,,  =  1  ^(1  -  .«-)■!  -f-  />U,  =  ^^    '  «  • 

17.  jyiscnssion  of  Darx-in'.y  proof.  Tiiese  results  ap-ee  with  each  other, 
and  witli  what  has  been  proven  in  another  way  ;  but  f.as  proof  of  the  faet 
that  //,  and  />  are  not  zero  nor  infinite  does  not  appear  to  be  entirely  satisfac- 
tory, and  it  is  esHoutial  that  this  property  of  />',  and  />  be  nnide  evident.  The 
ratio  /l„+2/^«+i  l>ecomes  1  -  i?  ;/-'  when  the  s.pmro  and  higher  powers  o^  n  ' 
are  neglected.  If  after  a  certain  value  of  n  .piantities  of  the  order  u  '  be 
neglected,  the  ratio  .l,.+,/^l„  beconu-s  unity.  Tlien,  foMowing  a  line  of  argu- 
ment similar  to  V  it  given,  it  would  appear  that 

„  =  A,  +  B.,{i  -^T'  (*) 

and,  by  a  similar  course  of  reasoning,  that 

dii/,h;  =  (',  +  A(l  -  x-y. 

These  results  do  not  agree  and  are  incorrect,  but  they  show  in  what  respect 


WHICH  I'liKsi'.NiH  iTsK.i.r  in  r.AiT.Acr.'H  KiNF/rtc  tiikohy  dk  tidkh. 


11.- 


tlif  iJifvioiiM  rciiMoniii^  Ih  wciik.      For,  Hupposo  tlint  tlin  liiiioiiiini  oxpiiiiHioii  of 
(1        .'•-)'  Im  written 

Tlit'ii,  if  the  intinito  Horien  can  !>r  writtt'ii  in  tlu>  form 

»  ==  J,  -f-  /;,  (1  -^■')i, 

wlioro  .l|  mill  /i\  ans  tiiiiti!  for  nil  viiliii's  of  /',  it  follows  tlmt 

•  .1, 


"■^i-l:- 


Now  for  over}'  fiiiito  valiio  of  ti  {^'i  \nni\^  ponitivo), 


Diu'win  liiiH  sliowii  tlmt 


A.    '  r. 


(b) 


l)nt  it  is  iiocossiiry  iilso  to  show  tlmt 


i 


V 


.1,  .1 

A,-  A, 


.1,        J, 


+1 


(0) 


is  tiiiitc. 

Both  miriifnitor  iiiid  (leiioininivtor  aro  zero  so  that  the  value  of  the  ([uoticiit 
retpiircs  iiivcstif^iitioii. 

The  tlisoussion  of  (a)  and  (i;)  shows  tlmt  in  (a)  A,  and  //.  do  n()t  have 
finite  uon-vauishiug  values  for  all  values  of  x. 

VII. 

18.  Ciitii's  to  hf  ti't'iiltd.  It  remains,  then,  to  exandne  the  other  cases 
included  iu  the  solution  obtained.  Airy  i)ointod  out  that  in  the  solution  of 
tlu!  form 

u=  liu,(.v)    I    V  (.'■), 

//  could  '■■>  determined  so  that  the  solution  would  l)e  suitable  for  the  oas(\  of 
a  sea  fonnin^  a  splu^rical  cap  and  extending,'  from  th(^  pole  to  au  arbitrary 
parallel  of  latitude.     Lord  Kelvin  jiointeil  out  that,  if  the  general  solution  were 


t\ 


.j;^ 


NPPI 


1  \(\  i,i\(i.     OX  ruv,  son  TioN  or  .\  ckiiiain  pirn;iiKNri.\i,  i-'.irAV)oN 

lit  hiitiil,  tli(>  two  rnnslinits  coulil  1m>  ili-ttMiiiiiic.l  so  iis  to  olttiiiii  ii  solution  snit- 
nl>l('  for  II  zoiiiil  s.'ii  Iviii^;  lictwct'ii  two  paiiillcls  of  liititiidc.  'I'lio  most  iiitri- 
fsliii^  ciiscs  to  l)t'  (Iciilt  with  a|)|M>iir  llien  to  lu'  III.'  foliowiiig  : — 

1.   r./,sr  (>f''t  s,ui  CDiyrliK/  the  wholo  <,irtli.      This  is  tin  ntseitlrMily  treated 

'2.   Cixti  of  It  si'it.  <'.vtyn>/iii>/j'i\>iii  //'.'  />"/'•  to  "  '.I'n't'.ii  /uwillcl  «/ /,itilii(t('. 

'i\.  (^iiKf.  of  it  ZKiKif  ,V(W  lioiniilr,!  hij  tiro  />iir,i//<''s  of  Itititti,!,'  on  oj>ji(Wite 
tili/eK  of  the  ei/ii>r/or  mol  rt/ini/ii/  ilisliiiil  from  it. 

I.  ('is,  ,if  (I  .dim/  st'i  hi, moll  <l  I'll  'iiiij  liro  piii'dllrls  of  /iif.'/infe  /i/lin/  in 
one  /ieiiii,y,,'ieri'. 

5.   Case  of  a  c,ii.,tl  lijiioj  nlomj  n  /hir,i//,i  of  hititiole. 


Cask  2. 

1'.».  /'o/iir  xeo.  If  tln>  s(>a  fxtciids  oiil.v  vO  ii  ^nv(-ii  iiariillcl  of  liitiliui<' 
from  111.'  i)ol(>,  it  is  •ii'.'cssiin  iliiit  tin'  nui  itlioniil  coiiiiioiicnt  of  llio  iii.)tioii 
sli.ml.l  v.iiiisli  for  111.'  corr.'sii.iii.liiij^  v.'iln.-  .  f  r,  llic  sin.'  of  lli.'  |>o!,'ir  .lisliini'i' 

of   llli'   I-.iiUIIiImI'V. 

'I'll. 'II.  as  in  Case  1.  .1  0.  and  111.-  condilioii  just  nam. '.I  ^iv.'s  for  tlic 
boundary  value  of  .'• 


fin 


it  //    „  0, 


Ij.'I.  lli.'ii.  ",  1>.'  Ill''  ■•ol;'liliid.'  of  111.'  sou 


111. 'I'll  lioiniilai'v.  an.l  sin  ",    --  k. 


(Ill        I'll 


II  ,   an.l   i.'s  II,       0 


,ln         'In 

(l.v       sin 


r^  (I,  f..r  l>  --  0, 


/!o;{u,)  ^  v'(«,)  ,  " ;/.'",(",)  i  vc/,);  -  o 


//-- 


«,V'(",)    I    '-JVC/,) 

",",'(«,)  f  'i"l("l) 


'I'liis  {^ivi's  i\\v  cxprossioii  for  u  at  anv  |>oinl  in  tlif  f.nni 


=  V(.'') 


,/.,V'('/,)    I    '2V('/,) 


A-'') 


In  jiailioular,  for  tlic  souIIi.m'u  li.Miiulan, 


'(«.)- 


{•.\r,) 


:i(i) 


(37) 


wiiuii  riiKSKNi'M  ri'si',1.1-  in  i,\i'i„\(r.  s  kinktic  ihko'iv  hk  rii>r.s. 


117 


(lil')  inul  {'M)  ^;iv(<  tlic  itnioniits  to  !i(>  iKiilcii  to  llio  tide  ili'tlnct'd  from  llic  t'i|ni- 
lil  riiim  llicory. 

Tlio  toliil  ti  !(•  ill  any  jioint  lli  'H  is 


f  =^  «,(,,■)    ,    /'.'.r  , 


iiiul  lit  tlic  souIIh'I'ii  Itoimdiiiv 


/'  =.-  II  («,)    j    /•'u^■ 


,:i8) 


( :!'•>) 


Ah  liiis  lu'cn  st.'itcd.  l-'crrcl's  ciilciiliitioiis  wcir  in.iilc  for  llic  series  in  wliicli   />' 
wtis  ji;iv('n  l>y  I  lie  rrhilioii 

/.'         /         0. 


'riiiMi,  from  (."t."))  it  npiK'ius  tiuil  tiic  tiilcs  imIi'iiIhIciI  by  I'crrcl  would  ix-  (Iiomo 
cxisliii};  on  !!  circiimpoiiM'  scm  lionndcd  by  ,1  |>nridl('l  of  latiludf  ,J~  ll^  wIhto 
a  =  Kin  l>  iinil  sulislics  the  riuialion 


/; 


iiV (11)   I   '2v  I") 
Cask  :{. 


('.Whf) 


■JO.  Si, I  <  ri,  iiiUiiii  ,  ijiKill  II  :>ii  Ih'tli  sill,  n  'l' ,  i/ii,il,ir.  Siipposi-  the  si'.'i  to 
cslcnd  i'i|n;dly  on  iiotli  siil<'s  of  llic-  ('i|UMtoi-.  ||ii>  lioiind.irit's  IxMnj;  jiarallcl^  of 
latitndc. 

'I'lio  condition  lliat  tlicrc  sliall  lie  no  motion  of  watiT  aloni;  llic  nu'ridiiin 
at  any  point  of  tlu>  nortin'rn  lionnilary  ij;iv(>s  one  relation  coniuu'linL;  .1  and  /•' ; 
Init  it  is  cli'Mr  1 1ml  the  ('oir(>spondiiii;'  eondilion  for  t  he  southern  iioinulMrv  L;ives 
exactly  the  same  relation  ;  so  th;it  one  of  the  constants  appears  to  l>e  arhilrarv. 
The  considerations  wiiich  applied  to  ( ';ise  i  apply  to  this  case.  The  s\  ninielrv 
of  tlic  motion  reipiires  that  theie  l>e  no  meridional  motion  of  the  water  at  the 
eijuator.      In  this  case,  also,  it  is  necessary  tlnit 

This  [fives  a  second  condition  In  means  of  which  th(>  remaininu  iirhilrarv  con- 
slant  may  he  determined. 
l''iom 

>i        .1   W    I    ",  l<'^;-,fi     I    li»,    i    V  {'I'D 

it  follows  th.d 


.In 
,10 


.1(1  —.»•■■)»  (f'    I    */,'lo^'.c    I     ^  //,)    I    /.\1        ,(■-)»«,■  -f  (1    -,/■••)»  v. 


M 


118  LINO.      ON  THE  HOLUTIDN  OF  \  CEISTAIN  OIFFEItESTIAL  EC^UATION 

Now 


£«- 


,7"-)4  V'J  =  0  ; 


r(l  -  .7r)l 


-^  «, 


0. 


since  «,  converges  for  .r  =  1  ;* 

_£[  (1 -*•-)'",' loj^  ■'■]  =  f'. 

siiici!  log  1=0  and  J^[(l  —  «;")■  "I'l  i*^  '""^'^• 
It  is  clear,  then,  tii.it 

vl  £l(l  -  •'.■:)^  <f'\  +  ^^  £  [(1  -~  ^')'  ">]  =  '>  • 

It  has  beea  seen  that 

4- [(1  -  •^")'  "I'l  i^  ^  *i"it<i  quantity,  say  f>  ; 

-J- 1(1  —  *'■)=  v'l  '**  '^  ^^"'''"  <l«'"itity,  say  f'  . 

(In  one  particular  ciiso  it  is  possible  that  a  niiglit  he  Z'^ro.)     Then   it  follows 

that  ^,,^^ 

Aa  +  ni>  -  0  .  (40) 

Returning  now  to  the  condition  first  stated,  let  «,  lie  tlie  colatitude  of  the 
boundiiry,  and  lot  sin  «,  ==  w,.     It  is  necessary  that 

since  cos  ";    '  0.     The  resulting  relation  between  A  and  />  takes  the  form 

+  yi|»;(«,)+ '^",(",)i-i- v(«,)  ,  'fv(«,)  =  «.  (12) 


I 


J 


♦  Hec  eciuiitiMii  ( b)  Section  VI.     The  I'xpaiiHiou  of  (1  —  x')l  fouviigcs  for  x  =  1. 


L^^ 


WHICH  PliESENTS  ITSELF  IN  I^VPLACE's  KINETIC  THEORY  OF  TIDES.  Hi) 


An  ctjimtion  for  //  is  obtainod  by  uliiiiiuntiu}^  A  and  />*  from  (22),  (40), 
and  (42).     Tlie  total  tidu  is 

Case  4. 

21.  Sfii  houiidi'il  hij  tiro  paralleli)  of  lat'dmle  mi  flw  .sdiiw  side  of  equator. 
Suppose  the  sea  to  l)e  bounded  on  the  north  and  south  bj-  parallels  of  latitude 
and  to  li(!  entirely  within  one  h(^niis|)here. 

It  is  necessary  tluit,  at  the  northern  and  southern  boundaries, 


dii       2 
dx       X 


n  -=  0 


Let  tii(!  boundaries  have  colatitudos   II ^,  11.^  (H^    r'  H.^),  and  let  sin  II ^  :=  «,,  sin//o 

Then  the  e(juations  tor  tiie  deterniiuation  of  A  and  li  are  similar  to  (42) 
and  are 

r  2  12  1 

A  I  <f'{u,)  +  "  c  («,)  +  ?/,'(«,)  log  «i  +       ".("i)  ^-       "i("i)log«i  I 
L  "\  "\  'h  J 

+  ii  I  ".' (",)  -I    'f  ".  («,)  1  +  V'(",)  +  \  V  («,)  =  0  , 

f  9  1  '^  1 

^I  I  f'K)  +  "  V'K)   i    "i'K)log«,  +       M,(//j  +      "i('A)log«2  I 


H43) 


As  before. 


A  \s^  (.'•)   I-  n,  {x)  log  ,r]  H    liu,  ix)  +  V  (.'•)  -  «  =  0  .  (22) 

Then,  eliminating  .4  and  II,  tiie  equation  for  «  is 

V'W  -f  "iv'-)log.r  ,•",(.'■)  ,  V(^0 

2  12  0  9 

f 'K)  -I         f  («i)  +  »<i'(«i)  !•>«  «i  +        "r(«i)    !     ,    "i("])log«n"i'(«i)  -!-  ,   »m("i).    ^A'"i)  ^-    ,    ''('/,) 

n  1  '2  2  2 

tf'(«,)  +  V'(«2)    -I      "l'(«-..)  log  «,  -4-  ".("i)   +  "l("-.01og«,,  1','("..)    -f  ,     ■"l(«2),    ^"(«2)   ^      „    V('/.,) 


9  19  2 

9  12  '' 


?<  .         (44) 


!  I 


120      UNO.   ON  THE  SOLUTION  OF  A  CEUTAIN  DIFFEHENTIAL  EQUATION 

(•14)  gives  the  value  of  u.     The  comp'ete  tidal  expvessiou  is 

u  -f  Kur . 
Till)  value  of  n  at  the  bouiulary  whose  colatitude  is  t\  is  given  by 

0.  .  y 


«?'(«,)  +  "  f(«,)   I-  »'(«l^l«g«l  +      "l(«l)  +  f  "■(«■)  •"R«H  «i'('«i)   I    ^t  "'^'''') 

9  12  2 

tf!'(«^)  +  "  y^K)  +  «'K)  log '-^i  r     "iK)-l-     '<i('A!)  log «:!.  "i'(«i)  I    ,  «i(«2) 


fC«,) 


1 


f '(«i)  -i-       "i('^i) 


,  V(«,) 
,  V'(«,) 


,(44A) 


9  a       1  2  '/  2  2 

f '(«2!* -t- "  f K) -f '«i'(«j)  log  '  +     "i(«2)  +     "if'-fo)  lo^'  -,«,'(«.)  +     ",{«o),  i"(«,)  +    I'KO 

//.,  //j  '/.»  '/.i  '/]  '/■■)  (X.t 


A  siii..lar  equatiou  may  be  obtaiuoil  for  the  evaluation  of  ii{a,). 

Case  5. 

22.  ('(inul  iif  ir'xjth  if]  hj'nnj  (ilomj  <i,  pii'dlli'.l  of  htt'itiil  e.  Suppose  tlio 
zonal  sea  to  narrow  down  to  a  canal  of  width  %l  having  as  its  northern  bound- 
ar\'  the  jiarallcl  of  latitude  \z  —  ". 

All  tliriH'  of  the  functions  in  the  expression  (22)  are  expressible  in  the 
neighborhood  of '/  =■■  H,  in  series  form.  Let  them  bo  expressed  in  this  man- 
ner and  let  '/  be  taken  so  small  that  powers  of  it  higlier  tlian  the  first  may  for 
pur|)ost's  of  calculation  be  neglected. 

TIh!  second  of  the  ecpxations  (i;])  then  becomes,  after  simplification  by 
means  of  the  first  one, 

r  2  9  2  1 

yi|f"(«i)-r       V''<'«i)  —    ""..  V'(«i)  -t-    ,   "i'('^i)    ■-     ,ih{a,) 
[_  <l\  'l\  "s  "■\ 

2  2  1 

+  ["i"(«i)  H       ".'(«i)  -  7,»<i(«i)llog'/i 
'/]  "i  J 


L 


J 


+  d  {I A  +  VI  li  +  n)  I   1  —  a}  =  0  ,     (4{i) 
where  I,  m,  n  may  be  easily  found  in  terms  of  «,. 


,li 


WHICH  I'KEHENTH  ITSELF  IN  LAPLACE'h 


?§ 


+ 


(>» 


<M     s" 


+ 


C^l  •  r 


KINETIC  THEOIIV  OF  TIDES.  121 


3-1 


+ 


n-i 


t^ 

■* 


•JT    ^     cc     5"      >— I 


i 
J 


(N 


+ 


1 

■^ 

;^ 

1—1 

■§ 

(MlS 

■      5^     5          _ 

0 

+ 

+ 

; 

a 

■-C 

^ 

*<" 

+ 

V        5 

"^ 

2: 

■r 

<0 

C 

^ 

^- 

tc 

f 

-V. ^ 

-:      " 

U) 

0^ 

■'  — 

C 

s 

s 

0 

■a 

/■-*, 

, 

*M 

g 

v£ 

1 

« 

B 

(N     jT 

1-H    "-■ 

^ 

g' 

+ 

+ 

0 

« 

5? 

( 

1 1 

s 

J3 

*<" 

^T" 

a 

a 

" 

^^ ■ 

^^-cC 

Sm 

V 

f-i     5 

•C 

i 

X 

^ 

to 

01     ^ 

+ 

^1  V 

1 

H 

^ 

^^ 

■"^ 

fs 

5<" 

^ 

^ 

^ 

M 

•««j 

(N  !> 

(M     ;?■ 

^ 

f* 

1 

' 

4- 

tc 

-j- 

1 

^ 

_0 

^r 

^ 

*s* 

+ 

u. 
n  »- 

y 

"u^ 

(>j   a"    «^^    5~ 


o^  I  a"      — 


^1   sT    cc 


e  tc 


tc 


■^     a 


tc 


+         + 


5a- 


+ 


+ 


V        + 


Oi 


t^"    5"    eo  i  a"      — - 


i- 


"J-  ^j- 


(Mi',r 


tc 


+ 


122  LINO.      ON  THE  SOLUTION  OF  A  CEUTAIN  DIFFERENTIAL  ECJUATION 

23.  Title  III  j)oint  dintant  d  from  hoamliiry  of  canal.  For  any  point 
within  till)  canal  distant  i)  from  the  northern  boundary  x  —  a^  -f  ii  y'\  ~  a^. 
This  siihstitntion  beinf^  made  and  the  product  </<)  beinj,'  neglected  the  result  is 


2  7 

f'(«i)  +  „  f('/) +  -«,(«,) 

3  , 


>  ".'(«.)  +  I  ".(«■)  ,  V'(«,)  +  -  V(«,) 


•'i  "■{  "■\  a,  a. 


f(«i) 


,   «,(«,) 


^'(«.) 


^  —  Wi  log  «, 


?», 


f'(«.)  +  -  «l(«l) 


.   '</(«.) 


,  ^^"(«.) 


+  S  I    1  -  «?   ^'(«.)  +  ,"[  ",(«,)  +  ^  ^(«,) 


",'('^.)  -f  I  '«>(«.)  ,    ^"(«.)  +  ~  i'(«,) 


'*1  "l  '*!  Ui  li^ 


V'(«l)   +   I  f  («l)  +   ■"■-  M,(«,) 
«1  «! 


.     "l'(«l)    +    „    "l(«l) 


"1  '*1  «i  «i 


+  u  I,  1  ~a^-<l 

I  —  111  log  '/, 

It  is  easy  to  show  that 


(48) 


III, 


I  -  III  log  a,  =  ,f-{a,)  +       ^."(«.)  -  ,;.  f '(«i)  +  A  f  («,) 

'*l  '*l  f^i 

"i  "■]  '-i 

2  4  4 

m  =  «i"'('/,)  +     ■  *«,"('/,)  —     „  >i\u,)   -f-    *   ,/,(,/,)  , 

n  =  V"'(«0  +  ^  V"(«0  -  f,  i"(«0  +   \  V(«,) . 
"i  «i  «i 


/ 


WHICH  rnKSKNTH  n.K.K  IN  ..placb's  kxne^c  THKonv  or  riOKS. 
luination  (48)  can  then  be  put  i..  the  form 


123 


f'K)    H^fK)   rj;«i(«.) 


.  «»'(«»)  +  «,"'(«'^ 


.>o  +  ^x«o<"/(^o  +  J«.i«o.<(.)  +  ^'''(«o 


«, 

2    ..„ 


+  '/ 


> 


f  («i) 


_  V^'(«.)  +  «,  "'^''') 


,<(«.). n«.)  fi+''^'  ''V'^^^' 


+ 


<r'(«,)  ^       "/«■) 


(49) 


.He  value  o.  .  at  th.  .lie  o.  th.  la.al  .  oUaiue^  h,  puttin.  .  =  .,  ^^-^eh 

U.e  .idth  n.ay  be  neglected  (40)  .  ^^l^^^;^  ;,  Uo  aimensions. 
.    coiuc•iae^vith  those  obtained  by  cousiueiing  the 

In  this  fase  the  equation  for  u  is 


^■K)  +  !  f  K)   '   «.  "'^"'^ 


«.'(«.)  -r  ,7  "■("') 


"l  '  ,.x        u/.,  \     1 


,    «,'(«,).    S^"(«.) 


(r>0) 


2      V    \         ^  ,1  (a)    »,"(«i).  ^^"("i) 


124  LINO.      ON  THE  KOL'J'J'ION  01'  A  CEUTAIN  DIFFE/iENTIAL  E(K'ATION 

As  bffore,  tlio  totiil  tide  is 

VIII. 


(51) 


SlMMAIlV   UK    HehULTS. 

25.  SinniiKiry  i>f  f-/V.  For  coiivtsiiieiit'o  of  icfoiuiico,  and  in  ordor  to 
riuuUr  tlio  resnltH  available  to  any  who  do  not  dosiro  to  follow  tiirongli  the 
proi'csscH  of  ohtainiiif,'  them,  it  has  been  thought  desirable  that  tliey  should 
b<'  restated  in  a  separate  section  which,  along  with  the  historical  sketch  in 
Section  11,  would  give  a  complete  account  of  the  state  of  the  i)roblem.  Sec- 
tion III  is  devoted  to  a  discussion  and  criticism  of  the  analytical  process  by 
means  of  v,hich  Laplace  obtained  his  valua  for  the  arbitrary  eonstaut  iu  his 
solution.  Objection  is  taken  to  tlu;  process  i-miiloyed  for  two  reasons.  Tu  the 
lirst  place,  the  reasoning  used  has  not  been  shown  to  be,  and  docis  not  appear 
to  1)6  strictly  accurate.  In  this  connection  it  may  be  said  that  iu  the  oxami- 
Uiition  of  the  iijjparent  inaccuracies  it  has  been  thought  sutlicient  to  indicate 
the  weaknesses  of  the  method  rather  than  go  into  a  minute  discussicm  of  them. 
The  modern  advances  in  the  theory  of  DilVerential  E.puitions  make  it  appear 
prol.abhi  that  matters  of  this  character  will  be  treated  ditlerently  in  future, 
in  Section  IV  the  complete  solution  of  the  e(puition  is  found,  the  expressions 
involved  being  infinitt!  series,  whose  r.'gions  of  convergence  are  large  enough 
to  m.ike  iH)ssil)le  the  treatment  of  all  cases  that  can  arise.  The  regicms  of 
convergence  of  the  series  tog(!tlier  with  certain  important  properties  of  the 
integrals  can  be  |)redieted  from  the  form  of  the  tupiation.  The  integral  fouiul 
is  more  general  than  that  previously  deiluced  involving  the  two  ari)itrary  con- 
stants. Th(!  series  used  by  Laplace  (inters  this  integral  as  a  part  of  it.  In 
the  derivation  of  the  integral  the  method  of  Laplace  as  given  in  the  Mecanique 
Celeste  is  made  clear.  In  the  closing  paragrai)hs  of  the  section  certain  pro])- 
<rties  of  the  integral  important  iu  the  applicatiou  of  the  physical  conditions 
are  deduciid. 

20.  S„ww,ini  of  V-  VII.  Section  V  deals  with  the  ai)p]icatious  of  the 
physical  conditions  to  the  determination  of  the  arbitrary  constants.  One  of 
the  constants  is  immediately  determined.  The  determination  of  the  other 
involves  greater  diflicultie.s.  The  analysis  on  which  i)revious  evaluations  have 
i)een  based  is  rej<'cted  and  replaced  by  a  determination  of  the  value  which 
appears  to  be  entirely  satisfactory.  The  condition  which  determines  this 
constant  is  the  condition  stated  by  Lord  Kelvin.  The  result  of  this  section 
a[)pears  to  be  a  complete  justification  of  Laplace's  series. 


WHICH  I'BEHENTH  ITSELF  IN  LAI'LACE'h  KINETIC  THEOUY  OK  TIDEH. 


I'ir. 


In  Section  VI  tlio  objoetiouH  tiikou  to  tlio  ])roof  proviouHly  f^ivon  in  thn 
(It'tormiimtiou  of  tlio  Hocond  arl)itniry  oonHtaiit  iiic  not  forth. 

The  hiHt  Sc'tion  contniuH  ilit^  diHCUHsion  of  five  iniportimt  cases  in  the 
tli(iory  of  tides.  The  arl)itiaiy  cojistants  in  the  (general  inte^'ial  of  tiu!  dirt'er- 
ential  equation  are  deteiniined  so  that  the  intemids  lepiesent  tiie  ti(hd  distnrli- 
nnce  in  those  cnses,  and  expressions  are  obtained  for  tlie  tidal  disturbance  at 
any  point  whatever,  and  at  certain  particular  points,  snch  as  points  on  the 
boundary.  Tiie  last  of  the  five  cases  treated  is  that  of  a  canal  lyinfi;  alonj^  a 
])arallel  of  latitu<le  and  would  apjcar  to  furiilHli  a  means  of  checking  tlie  same 
case  treated  by  Airy's  Canal  Theoiy  o(  Tides. 


ReFERENC'FS   TO   LlTERATUnE   OF   THE   PlIOltLFM. 


I 


Lai'LACE  :  Eecherches  sur  ))liisicur.i  j)oints  du  Systeme  du   Monde  ((Kuvr<!S, 
t.  ix). 
Des   oscillations  de  la  nier  et  dc  ratniosjdiere  (Mecaniquo  Celeste 
Livre  TV). 
AiiiY  :  Tides  and  Waves  (Encyclopedia  Metropolitana). 

On   a  controvcM-ted   point  in  Laplac(!'s  tiicory  of  tides  ('Philosophical 
Ma^'azine,  ()ctoi)er,  1H75). 
Kelvin:   Note  on  an  alleged  error  in   Ija])lace's  tlieory  of  tides  (Philosophical 

"*^af;azine,  Septemlu^r,  187')). 
Fekrel  :  Tidal  llescarches  (Ai.pi^ndix  to  tiie  Ignited  States  Coast  and  Geodetic 
Surv(!y  l{(!port,  1S71). 
On  a  controverted  point  in    Laplace's  tlujory  of  tides  (Philosophical 
jVEagazine,    Marcii,   l.S7(i,    also    (tould's    Astntnomical   Journal. 
Vols.    U   and    10,   and    Smithsonian    Miscellaneous    Collections, 
No.  »-l'i). 
Daiiwin  :  Tides  (Encyclopedia  JJritannica). 
Basset  :   Treatise  on  Hydrodynamics,  A'ol.  II. 
La.mb  :  Hydrodynamics,  second  edition. 


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